Weyl symmetry in $(D_4,D_4)$ conformal matter on a circle

Abstract

We study Weyl symmetry in quadrivalently glued 5-brane webs of rank $N$ $(D_4,D_4)$ conformal matter theories on a circle. We find that these theories all have affine $E_8$ Weyl symmetry in their brane webs, which indicates that they all have affine $E_8$ global symmetry. When $N\geq 2$, the theory has 64 different sets of affine $E_8$ invariant Coulomb branch parameters.